1. Field of the Invention
The invention generally relates to digital communications. In particular, the invention relates to filtering techniques in a receiver.
2. Description of the Related Art
The performance of digital communication systems can be measured in terms of Bit Error Rate (BER). Two common blocks found in digital receivers are: Decision Feedback Equalizers (DFEs) and Variable Gain Amplifiers (VGAs). Some features of both blocks will be discussed in this section.
A common type of nonlinearity encountered in practical amplifiers is compression, as depicted in FIG. 1. In FIG. 1, it can be seen that as input power increases, output power eventually saturates. This is referred to as gain compression. The 1 decibel (decision block) compression point (a commonly applied metric) refers to the point at which the gain of the amplifier drops by 1 dB below the small signal gain (the gain at very small input power levels).
Adaptive Filtering and Decision Feedback Equalizers
Adaptive filtering is a common and powerful function that finds use in a large variety of applications. One application is the basic communication problem, in which information is sent from one place to another. When an applied filter is used to compensate for the effects of the channel across which the information was sent, it is typically referred to as an equalizer.
One source of error in information transmission is intersymbol interference (ISI), which arises when a signal is sent across a dispersive channel. Dispersive channels tend to spread the energy of a transmitted signal out over time, so that both past and future symbols can interfere with the current symbol.
To further illustrate this point, consider a transmitted signal, t, which is sent across a dispersive channel with impulse response h. The received signal, x, is given by Equation 1.
                              x          k                =                                            ∑              n                        ⁢                                                  ⁢                                          h                n                            ⁢                              t                                  k                  -                  n                                                              =                                                    h                0                            ⁢                              t                k                                      +                                          ∑                                  n                  <                  0                                            ⁢                                                          ⁢                                                h                  n                                ⁢                                  t                                      k                    -                    n                                                                        +                                          ∑                                  n                  >                  0                                            ⁢                                                          ⁢                                                h                  n                                ⁢                                  t                                      k                    -                    n                                                                                                          Eq        .                                  ⁢        1            
The second term in Equation 1 arises from the precursor component of the channel impulse response, and models interference from future symbols with the current symbol. The third term in Equation 1 arises from the postcursor component of the channel impulse response, and models interference from previous symbols with the current symbol. Equalization techniques can be used to reduce or remove these components.
Oftentimes, prior knowledge of the channel characteristics is not known, making it difficult to define a relatively good, such as an optimum filter. To overcome this problem, filters are often made adaptive, allowing the filters to “learn” the channel characteristics.
Adaptive Transversal Filters
The adaptive transversal filter is a typical component in adaptive equalization applications, and is a well-understood non-recursive structure. It operates in the discrete time domain and has a finite impulse response (FIR). A generalized block diagram of the adaptive transversal filter is shown in FIG. 2.
For convenience, the input history and coefficients are expressed as vectors in Equations 2 and 3.
                                                        X              k                                                          =                              [                                  x                  k                                                                                        x                              k                -                1                                                          ⋯                                                                                x                                      k                    -                    N                                                  ]                            T                                                          Eq        .                                  ⁢        2                                                                    W              k                                                          =                              [                                  W                  ⁢                                                                          ⁢                                      0                    k                                                                                                          W              ⁢                                                          ⁢                              1                k                                                          ⋯                                                                                WN                  k                                ]                            T                                                          Eq        .                                  ⁢        3            
Coefficient adaptation is performed based on the desired response, dk, and the filter output, yk. dk is often a training (pilot) signal, which can be a copy of the transmitted sequence stored in the receiver, or the hard decisions of a Decision Feedback Equalizer (DFE). Commonly used adaptation algorithms attempt to minimize the mean-square error, E[εk2], where the error signal is given by Equation 4.
                              ɛ          k                =                                            d              k                        -                          y              k                                =                                    d              k                        -                                          W                k                T                            ⁢                              X                k                                                                        Eq        .                                  ⁢        4            
Equation 5 expands the square of the error signal.
                                                                        ɛ                k                2                            =                            ⁢                                                (                                                            i                      k                                        -                                                                  W                        k                        T                                            ⁢                                              X                        k                                                                              )                                2                                                                                        =                            ⁢                                                d                  k                  2                                +                                                      W                    k                    T                                    ⁢                                      X                    k                                    ⁢                                      X                    k                    T                                    ⁢                                      W                    k                                                  -                                  2                  ⁢                                      d                    k                                    ⁢                                      X                    k                    T                                    ⁢                                      W                    k                                                                                                          Eq        .                                  ⁢        5            
To produce a reasonably simplified expression for the mean-square error, several assumptions can be made: Wk is fixed; and Xk, dk, and εk are statistically wide-sense stationary. With these assumptions, the mean-square error is reduced as shown in Equation 6.
                              E          ⁡                      [                          ɛ              k              2                        ]                          =                              E            ⁡                          [                                                d                  ⁡                                      [                    k                    ]                                                  2                            ]                                +                                    W              T                        ⁢                          E              ⁡                              [                                                      X                    k                                    ⁢                                      X                    k                    T                                                  ]                                      ⁢            W                    -                      2            ⁢                                                  ⁢                          E              ⁡                              [                                                      d                    ⁡                                          [                      k                      ]                                                        ⁢                                      X                    k                    T                                                  ]                                      ⁢            W                                              Eq        .                                  ⁢        6            
From Equation 6, it is clear that the mean-square error is a quadratic function of the coefficient vector W. This quadratic function is referred to as the error surface, and it contains a global minimum at the optimal coefficient vector. In one embodiment, a task of the adaptation engine is to walk the coefficients down the error surface to a point as close as possible to the optimal solution.
There are a large variety of basic algorithms available to converge the coefficient vector to the optimal solution, including Newton's method, the steepest descent method, least-mean square (LMS), and recursive least squares (RLS). LMS is one of the most commonly used algorithms due to its ease of computation. LMS achieves its simplicity by approximating the mean-square error, E[εk2], with εk2, leading to the following coefficient update equation expressed in Equation 7.
                              W                      k            +            1                          =                              W            k                    +                                    μɛ              k                        ⁢                          X              k                                                          Eq        .                                  ⁢        7            
In Equation 7, μ is a step-size scalar that can be used to control convergence rate and steady-state accuracy.
In the above illustrative description, the filter and associated algorithms operate on real-valued data. The extension to complex-valued data and coefficients is well known in the art and is included in the scope of the present disclosure. Similarly, in the illustrated description, an optimal coefficient vector is typically chosen as the one that minimizes the mean square error between the filter output and the desired response.
Blind Equalization
When the desired response, dk, is unknown, adaptation should be done in blind mode. There are many algorithms capable of blindly converging for an adaptive filter, and they make use of higher-order statistics of the filter's input. Some prominent algorithms include Sato's algorithm and the Constant Modulus Algorithm (CMA).
Decision Feedback Equalizers
An alternative to the feedforward transversal filter, known as the Decision Feedback Equalizer (DFE), was originally proposed by Austin in 1967 and showed superior performance to its linear counterpart. It was later modified by George et al. (1971) to be adaptive. Adaptive DFEs typically use adaptive transversal filters in both feedforward and feedback roles (although the feedforward section is not mandatory), as shown in FIG. 3.
The role of the feedforward section is to reduce the precursor component of the ISI, while the feedback section reduces the postcursor component. In a traditional symbol-rate DFE, precursor and postcursor components spaced at integer multiples of the symbol period are corrected. For example, a DFE with N feedback taps can correct for postcursor components that occur at spacings of T, 2T, . . . , NT from the current symbol.
DFEs can be implemented in analog or in digital form. A digital implementation uses analog-to-digital conversion of the filter's input signal.
DFEs are often operated in decision-directed mode, which uses the output of the decision device as the desired signal. In this case, the error signal is given by the difference between the decision device's output and input. This is advantageous, as it does not require a training signal to converge the adaptation engine, although convergence can be more difficult. A block diagram for a decision-directed DFE is illustrated in FIG. 4.
In FIG. 4, a common error signal and adaptation engine are used to adapt both the feedforward and the feedback sections. The generation of this error signal can be challenging, as in an analog implementation, it is typical to sample and hold and then scale the soft decisions (input of the decision device) before subtracting them from the hard decisions (output of the decision device). This ensures that the delay through the decision device is accounted for, and also prevents the hard decisions from swamping the small signal level of the soft decisions.
Fractionally Spaced Equalizers
Fractionally Spaced Equalizers (FSEs) are transversal equalizers (used as a linear equalizer or the feedforward portion of a DFE) whose taps are spaced at some fraction of the symbol period. A typical choice is T/2 spacing, which allows correction of both the in-phase instant and the quadrature instant in the channel impulse response.
For an ideal, jitter-free sampling clock, equalization of anything but the ideal in-phase sampling instant provides no improvement in performance. However, when a realistic, jittered clock is considered, the true sampling instant slides around the ideal point. Because of this, there is advantage in providing equalization across more of the symbol period. In real-word systems, FSEs provide superior performance to symbol-rate equalizers.
RAM-DFEs store a list of feedback correction values in a table which is indexed by the pattern history. These have been used primarily in disk-drive applications. There does not appear to be a prescribed technique associated with RAM-DFEs for programming or indexing the table of coefficients to overcome BER degradations due to nonlinearities in an amplifier, such as a variable gain amplifier.
Yungsoo Kim et al., in “A Decision-Feedback Equalizer with Pattern-Dependent Feedback for Magnetic Recording Channels”, IEEE Transactions on Communications, Volume: 49, Issue: 1, pg. 9-13, January 2001, describes a RAM-DFE that attempts to address pattern dependent nonlinearities, but in complex and different way than the techniques disclosed herein. Kim's paper describes a method that is not based on feedforward information supplied by a preceding VGA, but rather solely on information internal to the DFE.
Many attempts have also been made on the transmit side to improve linearity. This has been done with analog feedback techniques (e.g., Cartesian feedback, polar feedback, dynamic biasing), digital feedback techniques (e.g., pre-distortion), and amplifier design (e.g., use of highly linear Class A structures).
The negative impact of nonlinearities on the Bit Error Rate (BER) of digital communication systems is well known. Several methods have already been developed to reduce the sensitivity of a digital communications system to nonlinearities in the signal path. These include, but are not limited to, predistortion, Cartesian feedback, and the design of composite amplifiers. Most of these solutions are targeted towards the transmit side of the link.
One relatively common situation found in a digital communication receiver is illustrated in FIG. 5. In FIG. 5, the VGA represents a variable gain amplifier, the AGC represents an automatic gain control block, and the DFE represents a decision feedback equalizer. Despite its apparent complexity, FIG. 5 is relatively simple to understand. The VGA amplifies the incoming RX signal to a desired level. The AGC feedback loop ensures the correct amount of gain is applied by the VGA. The DFE is a filter that helps remove inter-symbol interference (ISI) from the incoming signal.
The VGA can be a major source of nonlinearity in the receiver. One particularly common form of nonlinearity in a VGA is compression, which is often described using the 1 dB compression point (the signal power level at which the gain of the VGA drops by 1 dB).
Oftentimes, the incoming signal can have a large peak-to-average ratio (PAR), as a result of the modulation scheme, ISI, or some other phenomenon. To avoid degradations in BER associated with nonlinearities, the VGA is backed off from its 1 dB compression point. Generally speaking, one of two scenarios results: 1) the implementation of an overly power-hungry/complex VGA, or 2) an unacceptably high BER.